<?xml version="1.0"?>
<div class="add">

<b>set </b><font color="Maroon" title="c3">G</font> =  <a href="msinst_1.html#K4" title="MSINST_1:func.4">MSAlgCat</a> <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">A</font>;<br/>
<b>thus </b><a NAME="E1:13_1_1"/>
 <a href="msinst_1.html#K4" title="MSINST_1:func.4">MSAlgCat</a> <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">A</font> is <a href="altcat_1.html#V2" title="ALTCAT_1:attr.2">transitive</a>
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="msinst_1.html#K4" title="MSINST_1:func.4">MSAlgCat</a> <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">A</font> is <a href="altcat_1.html#V11" title="ALTCAT_1:attr.11">associative</a> &amp;  <a href="msinst_1.html#K4" title="MSINST_1:func.4">MSAlgCat</a> <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">A</font> is <a href="altcat_1.html#V12" title="ALTCAT_1:attr.12">with_units</a> )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">o1</font>, <font color="Maroon" title="c5">o2</font>, <font color="Maroon" title="c6">o3</font> be   <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <span class="p1">(<span class="default"><a href="msinst_1.html#K4" title="MSINST_1:func.4">MSAlgCat</a> <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">A</font></span>)</span>; <i><font color="Red">:: according to </font></i><a class="ref" href="altcat_1.html#D4" title="ALTCAT_1:def.4">ALTCAT_1:def 4</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c4">o1</font>,<font color="Maroon" title="c5">o2</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c5">o2</font>,<font color="Maroon" title="c6">o3</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or  not <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c4">o1</font>,<font color="Maroon" title="c6">o3</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )</span><br/>

<b>reconsider </b><font color="Maroon" title="c7">o1'</font> = <font color="Maroon" title="c4">o1</font>, <font color="Maroon" title="c8">o2'</font> = <font color="Maroon" title="c5">o2</font>, <font color="Maroon" title="c9">o3'</font> = <font color="Maroon" title="c6">o3</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="msinst_1.html#K2" title="MSINST_1:func.2">MSAlg_set</a> <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">A</font> <b>by </b><i><a class="ref" href="msinst_1.html#D4" target="_self" title="MSINST_1:def.4">Def4</a></i>;<br/>
<b>assume </b><a NAME="E2:13_1_1_1"/>
( <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c4">o1</font>,<font color="Maroon" title="c5">o2</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c5">o2</font>,<font color="Maroon" title="c6">o3</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> not <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c4">o1</font>,<font color="Maroon" title="c6">o3</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span><br/>

<a NAME="E3:13_1_1_1"/><b>then </b><i><font color="Green" title="E9">A1</font></i>: 
(  <a href="msinst_1.html#K3" title="MSINST_1:func.3">MSAlg_morph</a> <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c7">o1'</font>,<font color="Maroon" title="c8">o2'</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp;  <a href="msinst_1.html#K3" title="MSINST_1:func.3">MSAlg_morph</a> <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c8">o2'</font>,<font color="Maroon" title="c9">o3'</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 
<b>by </b><i><a class="ref" href="msinst_1.html#D4" target="_self" title="MSINST_1:def.4">Def4</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c10">t</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="msinst_1.html#K3" title="MSINST_1:func.3">MSAlg_morph</a> <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c7">o1'</font>,<font color="Maroon" title="c8">o2'</font>;<br/>
<b>consider </b><font color="Maroon" title="c11">s</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="msinst_1.html#K3" title="MSINST_1:func.3">MSAlg_morph</a> <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c8">o2'</font>,<font color="Maroon" title="c9">o3'</font>;<br/>
<b>consider </b><font color="Maroon" title="c12">M</font>, <font color="Maroon" title="c13">N</font> being  <a href="msualg_1.html#V4" title="MSUALG_1:attr.4">strict</a> <a href="msualg_6.html#V1" title="MSUALG_6:attr.1">feasible</a>  <a href="msualg_1.html#L3" title="MSUALG_1:struct.3">MSAlgebra</a> of <font color="Maroon" title="c1">S</font>, <font color="Maroon" title="c14">f</font> being   <a href="msualg_3.html#NM1" title="MSUALG_3:NM.1">ManySortedFunction</a> of <font color="Maroon" title="c12">M</font>,<font color="Maroon" title="c13">N</font><b> such that </b><br/><a NAME="E7:13_1_1_1"/><i><font color="Green" title="E10">A2</font></i>: 
( <font color="Maroon" title="c12">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">o1'</font> &amp; <font color="Maroon" title="c13">N</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">o2'</font> &amp; <font color="Maroon" title="c14">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">t</font> &amp; the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c12">M</font> <a href="pzfmisc1.html#R1" title="PZFMISC1:pred.1">is_transformable_to</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c13">N</font> &amp; <font color="Maroon" title="c14">f</font> <a href="msualg_3.html#R1" title="MSUALG_3:pred.1">is_homomorphism</a> <font color="Maroon" title="c12">M</font>,<font color="Maroon" title="c13">N</font> )
 <b>by </b><i><a class="txt" href="msinst_1.html#E3:13_1_1_1"><i><font color="Green" title="E9">A1</font></i></a>, <a class="ref" href="msinst_1.html#D3" target="_self" title="MSINST_1:def.3">Def3</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c15">M1</font>, <font color="Maroon" title="c16">N1</font> being  <a href="msualg_1.html#V4" title="MSUALG_1:attr.4">strict</a> <a href="msualg_6.html#V1" title="MSUALG_6:attr.1">feasible</a>  <a href="msualg_1.html#L3" title="MSUALG_1:struct.3">MSAlgebra</a> of <font color="Maroon" title="c1">S</font>, <font color="Maroon" title="c17">g</font> being   <a href="msualg_3.html#NM1" title="MSUALG_3:NM.1">ManySortedFunction</a> of <font color="Maroon" title="c15">M1</font>,<font color="Maroon" title="c16">N1</font><b> such that </b><br/><a NAME="E9:13_1_1_1"/><i><font color="Green" title="E11">A3</font></i>: 
( <font color="Maroon" title="c15">M1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">o2'</font> &amp; <font color="Maroon" title="c16">N1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">o3'</font> &amp; <font color="Maroon" title="c17">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">s</font> &amp; the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c15">M1</font> <a href="pzfmisc1.html#R1" title="PZFMISC1:pred.1">is_transformable_to</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c16">N1</font> &amp; <font color="Maroon" title="c17">g</font> <a href="msualg_3.html#R1" title="MSUALG_3:pred.1">is_homomorphism</a> <font color="Maroon" title="c15">M1</font>,<font color="Maroon" title="c16">N1</font> )
 <b>by </b><i><a class="txt" href="msinst_1.html#E3:13_1_1_1"><i><font color="Green" title="E9">A1</font></i></a>, <a class="ref" href="msinst_1.html#D3" target="_self" title="MSINST_1:def.3">Def3</a></i>;<br/>
<a NAME="E10:13_1_1_1"/><i><font color="Green" title="E12">A4</font></i>: 
the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c12">M</font> <a href="pzfmisc1.html#R1" title="PZFMISC1:pred.1">is_transformable_to</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c16">N1</font>
 
<b>by </b><i><a class="txt" href="msinst_1.html#E7:13_1_1_1"><i><font color="Green" title="E10">A2</font></i></a>, <a class="txt" href="msinst_1.html#E9:13_1_1_1"><i><font color="Green" title="E11">A3</font></i></a>, <a class="ref" href="autalg_1.html#T11" title="AUTALG_1:th.11">AUTALG_1:11</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c18">g'</font> = <font color="Maroon" title="c17">g</font> as   <a href="msualg_3.html#NM1" title="MSUALG_3:NM.1">ManySortedFunction</a> of <font color="Maroon" title="c13">N</font>,<font color="Maroon" title="c16">N1</font> <b>by </b><i><a class="txt" href="msinst_1.html#E7:13_1_1_1"><i><font color="Green" title="E10">A2</font></i></a>, <a class="txt" href="msinst_1.html#E9:13_1_1_1"><i><font color="Green" title="E11">A3</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c19">GF</font> being   <a href="msualg_3.html#NM1" title="MSUALG_3:NM.1">ManySortedFunction</a> of <font color="Maroon" title="c12">M</font>,<font color="Maroon" title="c16">N1</font><b> such that </b><br/><a NAME="E13:13_1_1_1"/><i><font color="Green" title="E13">A5</font></i>: 
( <font color="Maroon" title="c19">GF</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">g'</font> <a href="pboole.html#K11" title="PBOOLE:func.11">**</a> <font color="Maroon" title="c14">f</font> &amp; <font color="Maroon" title="c19">GF</font> <a href="msualg_3.html#R1" title="MSUALG_3:pred.1">is_homomorphism</a> <font color="Maroon" title="c12">M</font>,<font color="Maroon" title="c16">N1</font> )
 <b>by </b><i><a class="txt" href="msinst_1.html#E7:13_1_1_1"><i><font color="Green" title="E10">A2</font></i></a>, <a class="txt" href="msinst_1.html#E9:13_1_1_1"><i><font color="Green" title="E11">A3</font></i></a>, <a class="ref" href="msinst_1.html#T5" target="_self" title="MSINST_1:th.5">Th5</a></i>;<br/>
<a NAME="E14:13_1_1_1"/>
<font color="Maroon" title="c19">GF</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="msinst_1.html#K3" title="MSINST_1:func.3">MSAlg_morph</a> <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c7">o1'</font>,<font color="Maroon" title="c9">o3'</font>
 
<b>by </b><i><a class="txt" href="msinst_1.html#E7:13_1_1_1"><i><font color="Green" title="E10">A2</font></i></a>, <a class="txt" href="msinst_1.html#E9:13_1_1_1"><i><font color="Green" title="E11">A3</font></i></a>, <a class="txt" href="msinst_1.html#E10:13_1_1_1"><i><font color="Green" title="E12">A4</font></i></a>, <a class="txt" href="msinst_1.html#E13:13_1_1_1"><i><font color="Green" title="E13">A5</font></i></a>, <a class="ref" href="msinst_1.html#D3" target="_self" title="MSINST_1:def.3">Def3</a></i>;<br/>
<b>hence </b><a NAME="E15:13_1_1_1"/>
<span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c4">o1</font>,<font color="Maroon" title="c6">o3</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>by </b><i><a class="ref" href="msinst_1.html#D4" target="_self" title="MSINST_1:def.4">Def4</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>

<b>set </b><font color="Maroon" title="c4">M</font> =  <a href="msinst_1.html#K4" title="MSINST_1:func.4">MSAlgCat</a> <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">A</font>;<br/>
<b>set </b><font color="Maroon" title="c5">GM</font> = the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <span class="p1">(<span class="default"><a href="msinst_1.html#K4" title="MSINST_1:func.4">MSAlgCat</a> <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">A</font></span>)</span>;<br/>
<b>set </b><font color="Maroon" title="c6">C</font> = the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <span class="p1">(<span class="default"><a href="msinst_1.html#K4" title="MSINST_1:func.4">MSAlgCat</a> <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">A</font></span>)</span>;<br/>
<b>thus </b><a NAME="E2:13_1_1"/>
the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <span class="p1">(<span class="default"><a href="msinst_1.html#K4" title="MSINST_1:func.4">MSAlgCat</a> <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">A</font></span>)</span> is <a href="altcat_1.html#V3" title="ALTCAT_1:attr.3">associative</a>
  <i><font color="Red">:: according to </font></i><a class="ref" href="altcat_1.html#D17" title="ALTCAT_1:def.17">ALTCAT_1:def 17</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="msinst_1.html#K4" title="MSINST_1:func.4">MSAlgCat</a> <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">A</font> is <a href="altcat_1.html#V12" title="ALTCAT_1:attr.12">with_units</a></span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c7">i</font>, <font color="Maroon" title="c8">j</font>, <font color="Maroon" title="c9">k</font>, <font color="Maroon" title="c10">l</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="msinst_1.html#K4" title="MSINST_1:func.4">MSAlgCat</a> <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">A</font></span>)</span>; <i><font color="Red">:: according to </font></i><a class="ref" href="altcat_1.html#D7" title="ALTCAT_1:def.7">ALTCAT_1:def 7</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   holds <br/>(  not <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <span class="p1">(<span class="default"><a href="msinst_1.html#K4" title="MSINST_1:func.4">MSAlgCat</a> <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">A</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c7">i</font>,<font color="Maroon" title="c8">j</font> or  not <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <span class="p1">(<span class="default"><a href="msinst_1.html#K4" title="MSINST_1:func.4">MSAlgCat</a> <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">A</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c8">j</font>,<font color="Maroon" title="c9">k</font> or  not <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <span class="p1">(<span class="default"><a href="msinst_1.html#K4" title="MSINST_1:func.4">MSAlgCat</a> <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">A</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c9">k</font>,<font color="Maroon" title="c10">l</font> or <span class="p1">(<span class="default">the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <span class="p2">(<span class="default"><a href="msinst_1.html#K4" title="MSINST_1:func.4">MSAlgCat</a> <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">A</font></span>)</span> <a href="altcat_1.html#K4" title="ALTCAT_1:func.4">.</a> <font color="Maroon" title="c7">i</font>,<font color="Maroon" title="c9">k</font>,<font color="Maroon" title="c10">l</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Olive">b<sub>3</sub></font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default">the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <span class="p3">(<span class="default"><a href="msinst_1.html#K4" title="MSINST_1:func.4">MSAlgCat</a> <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">A</font></span>)</span> <a href="altcat_1.html#K4" title="ALTCAT_1:func.4">.</a> <font color="Maroon" title="c7">i</font>,<font color="Maroon" title="c8">j</font>,<font color="Maroon" title="c9">k</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>1</sub></font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <span class="p2">(<span class="default"><a href="msinst_1.html#K4" title="MSINST_1:func.4">MSAlgCat</a> <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">A</font></span>)</span> <a href="altcat_1.html#K4" title="ALTCAT_1:func.4">.</a> <font color="Maroon" title="c7">i</font>,<font color="Maroon" title="c8">j</font>,<font color="Maroon" title="c10">l</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <span class="p3">(<span class="default"><a href="msinst_1.html#K4" title="MSINST_1:func.4">MSAlgCat</a> <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">A</font></span>)</span> <a href="altcat_1.html#K4" title="ALTCAT_1:func.4">.</a> <font color="Maroon" title="c8">j</font>,<font color="Maroon" title="c9">k</font>,<font color="Maroon" title="c10">l</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Olive">b<sub>3</sub></font>,<font color="Olive">b<sub>2</sub></font></span>)</span>,<font color="Olive">b<sub>1</sub></font> )</span><br/>

<b>reconsider </b><font color="Maroon" title="c11">i'</font> = <font color="Maroon" title="c7">i</font>, <font color="Maroon" title="c12">j'</font> = <font color="Maroon" title="c8">j</font>, <font color="Maroon" title="c13">k'</font> = <font color="Maroon" title="c9">k</font>, <font color="Maroon" title="c14">l'</font> = <font color="Maroon" title="c10">l</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="msinst_1.html#K2" title="MSINST_1:func.2">MSAlg_set</a> <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">A</font> <b>by </b><i><a class="ref" href="msinst_1.html#D4" target="_self" title="MSINST_1:def.4">Def4</a></i>;<br/>
<b>let </b><font color="Maroon" title="c15">f</font>, <font color="Maroon" title="c16">g</font>, <font color="Maroon" title="c17">h</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c15">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <span class="p1">(<span class="default"><a href="msinst_1.html#K4" title="MSINST_1:func.4">MSAlgCat</a> <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">A</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c7">i</font>,<font color="Maroon" title="c8">j</font> or  not <font color="Maroon" title="c16">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <span class="p1">(<span class="default"><a href="msinst_1.html#K4" title="MSINST_1:func.4">MSAlgCat</a> <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">A</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c8">j</font>,<font color="Maroon" title="c9">k</font> or  not <font color="Maroon" title="c17">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <span class="p1">(<span class="default"><a href="msinst_1.html#K4" title="MSINST_1:func.4">MSAlgCat</a> <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">A</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c9">k</font>,<font color="Maroon" title="c10">l</font> or <span class="p1">(<span class="default">the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <span class="p2">(<span class="default"><a href="msinst_1.html#K4" title="MSINST_1:func.4">MSAlgCat</a> <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">A</font></span>)</span> <a href="altcat_1.html#K4" title="ALTCAT_1:func.4">.</a> <font color="Maroon" title="c7">i</font>,<font color="Maroon" title="c9">k</font>,<font color="Maroon" title="c10">l</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c17">h</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default">the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <span class="p3">(<span class="default"><a href="msinst_1.html#K4" title="MSINST_1:func.4">MSAlgCat</a> <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">A</font></span>)</span> <a href="altcat_1.html#K4" title="ALTCAT_1:func.4">.</a> <font color="Maroon" title="c7">i</font>,<font color="Maroon" title="c8">j</font>,<font color="Maroon" title="c9">k</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c16">g</font>,<font color="Maroon" title="c15">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <span class="p2">(<span class="default"><a href="msinst_1.html#K4" title="MSINST_1:func.4">MSAlgCat</a> <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">A</font></span>)</span> <a href="altcat_1.html#K4" title="ALTCAT_1:func.4">.</a> <font color="Maroon" title="c7">i</font>,<font color="Maroon" title="c8">j</font>,<font color="Maroon" title="c10">l</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <span class="p3">(<span class="default"><a href="msinst_1.html#K4" title="MSINST_1:func.4">MSAlgCat</a> <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">A</font></span>)</span> <a href="altcat_1.html#K4" title="ALTCAT_1:func.4">.</a> <font color="Maroon" title="c8">j</font>,<font color="Maroon" title="c9">k</font>,<font color="Maroon" title="c10">l</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c17">h</font>,<font color="Maroon" title="c16">g</font></span>)</span>,<font color="Maroon" title="c15">f</font> )</span><br/>

<b>assume </b><a NAME="E2:13_1_1_2"/><i><font color="Green" title="E9">A6</font></i>: 
( <font color="Maroon" title="c15">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <span class="p1">(<span class="default"><a href="msinst_1.html#K4" title="MSINST_1:func.4">MSAlgCat</a> <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">A</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c7">i</font>,<font color="Maroon" title="c8">j</font> &amp; <font color="Maroon" title="c16">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <span class="p1">(<span class="default"><a href="msinst_1.html#K4" title="MSINST_1:func.4">MSAlgCat</a> <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">A</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c8">j</font>,<font color="Maroon" title="c9">k</font> &amp; <font color="Maroon" title="c17">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <span class="p1">(<span class="default"><a href="msinst_1.html#K4" title="MSINST_1:func.4">MSAlgCat</a> <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">A</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c9">k</font>,<font color="Maroon" title="c10">l</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default">the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <span class="p2">(<span class="default"><a href="msinst_1.html#K4" title="MSINST_1:func.4">MSAlgCat</a> <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">A</font></span>)</span> <a href="altcat_1.html#K4" title="ALTCAT_1:func.4">.</a> <font color="Maroon" title="c7">i</font>,<font color="Maroon" title="c9">k</font>,<font color="Maroon" title="c10">l</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c17">h</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default">the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <span class="p3">(<span class="default"><a href="msinst_1.html#K4" title="MSINST_1:func.4">MSAlgCat</a> <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">A</font></span>)</span> <a href="altcat_1.html#K4" title="ALTCAT_1:func.4">.</a> <font color="Maroon" title="c7">i</font>,<font color="Maroon" title="c8">j</font>,<font color="Maroon" title="c9">k</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c16">g</font>,<font color="Maroon" title="c15">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <span class="p2">(<span class="default"><a href="msinst_1.html#K4" title="MSINST_1:func.4">MSAlgCat</a> <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">A</font></span>)</span> <a href="altcat_1.html#K4" title="ALTCAT_1:func.4">.</a> <font color="Maroon" title="c7">i</font>,<font color="Maroon" title="c8">j</font>,<font color="Maroon" title="c10">l</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <span class="p3">(<span class="default"><a href="msinst_1.html#K4" title="MSINST_1:func.4">MSAlgCat</a> <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">A</font></span>)</span> <a href="altcat_1.html#K4" title="ALTCAT_1:func.4">.</a> <font color="Maroon" title="c8">j</font>,<font color="Maroon" title="c9">k</font>,<font color="Maroon" title="c10">l</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c17">h</font>,<font color="Maroon" title="c16">g</font></span>)</span>,<font color="Maroon" title="c15">f</font></span><br/>

<a NAME="E3:13_1_1_2"/><b>then </b><i><font color="Green" title="E10">A7</font></i>: 
( <font color="Maroon" title="c15">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="msinst_1.html#K3" title="MSINST_1:func.3">MSAlg_morph</a> <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c11">i'</font>,<font color="Maroon" title="c12">j'</font> &amp; <font color="Maroon" title="c16">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="msinst_1.html#K3" title="MSINST_1:func.3">MSAlg_morph</a> <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c12">j'</font>,<font color="Maroon" title="c13">k'</font> &amp; <font color="Maroon" title="c17">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="msinst_1.html#K3" title="MSINST_1:func.3">MSAlg_morph</a> <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c13">k'</font>,<font color="Maroon" title="c14">l'</font> )
 
<b>by </b><i><a class="ref" href="msinst_1.html#D4" target="_self" title="MSINST_1:def.4">Def4</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c18">M1</font>, <font color="Maroon" title="c19">N1</font> being  <a href="msualg_1.html#V4" title="MSUALG_1:attr.4">strict</a> <a href="msualg_6.html#V1" title="MSUALG_6:attr.1">feasible</a>  <a href="msualg_1.html#L3" title="MSUALG_1:struct.3">MSAlgebra</a> of <font color="Maroon" title="c1">S</font>, <font color="Maroon" title="c20">F</font> being   <a href="msualg_3.html#NM1" title="MSUALG_3:NM.1">ManySortedFunction</a> of <font color="Maroon" title="c18">M1</font>,<font color="Maroon" title="c19">N1</font><b> such that </b><br/><a NAME="E5:13_1_1_2"/><i><font color="Green" title="E11">A8</font></i>: 
( <font color="Maroon" title="c18">M1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">i'</font> &amp; <font color="Maroon" title="c19">N1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">j'</font> &amp; <font color="Maroon" title="c15">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">F</font> &amp; the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c18">M1</font> <a href="pzfmisc1.html#R1" title="PZFMISC1:pred.1">is_transformable_to</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c19">N1</font> &amp; <font color="Maroon" title="c20">F</font> <a href="msualg_3.html#R1" title="MSUALG_3:pred.1">is_homomorphism</a> <font color="Maroon" title="c18">M1</font>,<font color="Maroon" title="c19">N1</font> )
 <b>by </b><i><a class="ref" href="msinst_1.html#D3" target="_self" title="MSINST_1:def.3">Def3</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c21">M2</font>, <font color="Maroon" title="c22">N2</font> being  <a href="msualg_1.html#V4" title="MSUALG_1:attr.4">strict</a> <a href="msualg_6.html#V1" title="MSUALG_6:attr.1">feasible</a>  <a href="msualg_1.html#L3" title="MSUALG_1:struct.3">MSAlgebra</a> of <font color="Maroon" title="c1">S</font>, <font color="Maroon" title="c23">G</font> being   <a href="msualg_3.html#NM1" title="MSUALG_3:NM.1">ManySortedFunction</a> of <font color="Maroon" title="c21">M2</font>,<font color="Maroon" title="c22">N2</font><b> such that </b><br/><a NAME="E7:13_1_1_2"/><i><font color="Green" title="E12">A9</font></i>: 
( <font color="Maroon" title="c21">M2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">j'</font> &amp; <font color="Maroon" title="c22">N2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">k'</font> &amp; <font color="Maroon" title="c16">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">G</font> &amp; the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c21">M2</font> <a href="pzfmisc1.html#R1" title="PZFMISC1:pred.1">is_transformable_to</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c22">N2</font> &amp; <font color="Maroon" title="c23">G</font> <a href="msualg_3.html#R1" title="MSUALG_3:pred.1">is_homomorphism</a> <font color="Maroon" title="c21">M2</font>,<font color="Maroon" title="c22">N2</font> )
 <b>by </b><i><a class="txt" href="msinst_1.html#E3:13_1_1_2"><i><font color="Green" title="E10">A7</font></i></a>, <a class="ref" href="msinst_1.html#D3" target="_self" title="MSINST_1:def.3">Def3</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c24">M3</font>, <font color="Maroon" title="c25">N3</font> being  <a href="msualg_1.html#V4" title="MSUALG_1:attr.4">strict</a> <a href="msualg_6.html#V1" title="MSUALG_6:attr.1">feasible</a>  <a href="msualg_1.html#L3" title="MSUALG_1:struct.3">MSAlgebra</a> of <font color="Maroon" title="c1">S</font>, <font color="Maroon" title="c26">H</font> being   <a href="msualg_3.html#NM1" title="MSUALG_3:NM.1">ManySortedFunction</a> of <font color="Maroon" title="c24">M3</font>,<font color="Maroon" title="c25">N3</font><b> such that </b><br/><a NAME="E9:13_1_1_2"/><i><font color="Green" title="E13">A10</font></i>: 
( <font color="Maroon" title="c24">M3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">k'</font> &amp; <font color="Maroon" title="c25">N3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">l'</font> &amp; <font color="Maroon" title="c17">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c26">H</font> &amp; the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c24">M3</font> <a href="pzfmisc1.html#R1" title="PZFMISC1:pred.1">is_transformable_to</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c25">N3</font> &amp; <font color="Maroon" title="c26">H</font> <a href="msualg_3.html#R1" title="MSUALG_3:pred.1">is_homomorphism</a> <font color="Maroon" title="c24">M3</font>,<font color="Maroon" title="c25">N3</font> )
 <b>by </b><i><a class="txt" href="msinst_1.html#E3:13_1_1_2"><i><font color="Green" title="E10">A7</font></i></a>, <a class="ref" href="msinst_1.html#D3" target="_self" title="MSINST_1:def.3">Def3</a></i>;<br/>
<a NAME="E10:13_1_1_2"/><i><font color="Green" title="E14">A11</font></i>: 
the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c18">M1</font> <a href="pzfmisc1.html#R1" title="PZFMISC1:pred.1">is_transformable_to</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c24">M3</font>
 
<b>by </b><i><a class="txt" href="msinst_1.html#E5:13_1_1_2"><i><font color="Green" title="E11">A8</font></i></a>, <a class="txt" href="msinst_1.html#E7:13_1_1_2"><i><font color="Green" title="E12">A9</font></i></a>, <a class="txt" href="msinst_1.html#E9:13_1_1_2"><i><font color="Green" title="E13">A10</font></i></a>, <a class="ref" href="autalg_1.html#T11" title="AUTALG_1:th.11">AUTALG_1:11</a></i>;<br/>
<a NAME="E11:13_1_1_2"/><i><font color="Green" title="E15">A12</font></i>: 
the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c21">M2</font> <a href="pzfmisc1.html#R1" title="PZFMISC1:pred.1">is_transformable_to</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c25">N3</font>
 
<b>by </b><i><a class="txt" href="msinst_1.html#E7:13_1_1_2"><i><font color="Green" title="E12">A9</font></i></a>, <a class="txt" href="msinst_1.html#E9:13_1_1_2"><i><font color="Green" title="E13">A10</font></i></a>, <a class="ref" href="autalg_1.html#T11" title="AUTALG_1:th.11">AUTALG_1:11</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c27">G'</font> = <font color="Maroon" title="c23">G</font> as   <a href="msualg_3.html#NM1" title="MSUALG_3:NM.1">ManySortedFunction</a> of <font color="Maroon" title="c21">M2</font>,<font color="Maroon" title="c24">M3</font> <b>by </b><i><a class="txt" href="msinst_1.html#E7:13_1_1_2"><i><font color="Green" title="E12">A9</font></i></a>, <a class="txt" href="msinst_1.html#E9:13_1_1_2"><i><font color="Green" title="E13">A10</font></i></a></i>;<br/>
<a NAME="E13:13_1_1_2"/><i><font color="Green" title="E16">A13</font></i>: 
<span class="p1">(<span class="default">the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <span class="p2">(<span class="default"><a href="msinst_1.html#K4" title="MSINST_1:func.4">MSAlgCat</a> <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">A</font></span>)</span> <a href="altcat_1.html#K4" title="ALTCAT_1:func.4">.</a> <font color="Maroon" title="c7">i</font>,<font color="Maroon" title="c8">j</font>,<font color="Maroon" title="c9">k</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c16">g</font>,<font color="Maroon" title="c15">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">G</font> <a href="pboole.html#K11" title="PBOOLE:func.11">**</a> <font color="Maroon" title="c20">F</font>
 
<b>by </b><i><a class="txt" href="msinst_1.html#E2:13_1_1_2"><i><font color="Green" title="E9">A6</font></i></a>, <a class="txt" href="msinst_1.html#E5:13_1_1_2"><i><font color="Green" title="E11">A8</font></i></a>, <a class="txt" href="msinst_1.html#E7:13_1_1_2"><i><font color="Green" title="E12">A9</font></i></a>, <a class="ref" href="msinst_1.html#D4" target="_self" title="MSINST_1:def.4">Def4</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c28">GF</font> being   <a href="msualg_3.html#NM1" title="MSUALG_3:NM.1">ManySortedFunction</a> of <font color="Maroon" title="c18">M1</font>,<font color="Maroon" title="c24">M3</font><b> such that </b><br/><a NAME="E15:13_1_1_2"/><i><font color="Green" title="E17">A14</font></i>: 
( <font color="Maroon" title="c28">GF</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c27">G'</font> <a href="pboole.html#K11" title="PBOOLE:func.11">**</a> <font color="Maroon" title="c20">F</font> &amp; <font color="Maroon" title="c28">GF</font> <a href="msualg_3.html#R1" title="MSUALG_3:pred.1">is_homomorphism</a> <font color="Maroon" title="c18">M1</font>,<font color="Maroon" title="c24">M3</font> )
 <b>by </b><i><a class="txt" href="msinst_1.html#E5:13_1_1_2"><i><font color="Green" title="E11">A8</font></i></a>, <a class="txt" href="msinst_1.html#E7:13_1_1_2"><i><font color="Green" title="E12">A9</font></i></a>, <a class="txt" href="msinst_1.html#E9:13_1_1_2"><i><font color="Green" title="E13">A10</font></i></a>, <a class="ref" href="msinst_1.html#T5" target="_self" title="MSINST_1:th.5">Th5</a></i>;<br/>
<a NAME="E16:13_1_1_2"/>
<font color="Maroon" title="c27">G'</font> <a href="pboole.html#K11" title="PBOOLE:func.11">**</a> <font color="Maroon" title="c20">F</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="msinst_1.html#K3" title="MSINST_1:func.3">MSAlg_morph</a> <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c11">i'</font>,<font color="Maroon" title="c13">k'</font>
 
<b>by </b><i><a class="txt" href="msinst_1.html#E5:13_1_1_2"><i><font color="Green" title="E11">A8</font></i></a>, <a class="txt" href="msinst_1.html#E9:13_1_1_2"><i><font color="Green" title="E13">A10</font></i></a>, <a class="txt" href="msinst_1.html#E10:13_1_1_2"><i><font color="Green" title="E14">A11</font></i></a>, <a class="txt" href="msinst_1.html#E15:13_1_1_2"><i><font color="Green" title="E17">A14</font></i></a>, <a class="ref" href="msinst_1.html#D3" target="_self" title="MSINST_1:def.3">Def3</a></i>;<br/>
<a NAME="E17:13_1_1_2"/><b>then </b>
<font color="Maroon" title="c28">GF</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <span class="p1">(<span class="default"><a href="msinst_1.html#K4" title="MSINST_1:func.4">MSAlgCat</a> <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">A</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c7">i</font>,<font color="Maroon" title="c9">k</font>
 
<b>by </b><i><a class="txt" href="msinst_1.html#E15:13_1_1_2"><i><font color="Green" title="E17">A14</font></i></a>, <a class="ref" href="msinst_1.html#D4" target="_self" title="MSINST_1:def.4">Def4</a></i>;<br/>
<a NAME="E18:13_1_1_2"/><b>then </b><i><font color="Green" title="E18">A15</font></i>: 
<span class="p1">(<span class="default">the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <span class="p2">(<span class="default"><a href="msinst_1.html#K4" title="MSINST_1:func.4">MSAlgCat</a> <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">A</font></span>)</span> <a href="altcat_1.html#K4" title="ALTCAT_1:func.4">.</a> <font color="Maroon" title="c7">i</font>,<font color="Maroon" title="c9">k</font>,<font color="Maroon" title="c10">l</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c26">H</font>,<font color="Maroon" title="c28">GF</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c26">H</font> <a href="pboole.html#K11" title="PBOOLE:func.11">**</a> <font color="Maroon" title="c28">GF</font>
 
<b>by </b><i><a class="txt" href="msinst_1.html#E2:13_1_1_2"><i><font color="Green" title="E9">A6</font></i></a>, <a class="txt" href="msinst_1.html#E9:13_1_1_2"><i><font color="Green" title="E13">A10</font></i></a>, <a class="ref" href="msinst_1.html#D4" target="_self" title="MSINST_1:def.4">Def4</a></i>;<br/>
<a NAME="E19:13_1_1_2"/><i><font color="Green" title="E19">A16</font></i>: 
<span class="p1">(<span class="default">the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <span class="p2">(<span class="default"><a href="msinst_1.html#K4" title="MSINST_1:func.4">MSAlgCat</a> <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">A</font></span>)</span> <a href="altcat_1.html#K4" title="ALTCAT_1:func.4">.</a> <font color="Maroon" title="c8">j</font>,<font color="Maroon" title="c9">k</font>,<font color="Maroon" title="c10">l</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c26">H</font>,<font color="Maroon" title="c23">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c26">H</font> <a href="pboole.html#K11" title="PBOOLE:func.11">**</a> <font color="Maroon" title="c27">G'</font>
 
<b>by </b><i><a class="txt" href="msinst_1.html#E2:13_1_1_2"><i><font color="Green" title="E9">A6</font></i></a>, <a class="txt" href="msinst_1.html#E7:13_1_1_2"><i><font color="Green" title="E12">A9</font></i></a>, <a class="txt" href="msinst_1.html#E9:13_1_1_2"><i><font color="Green" title="E13">A10</font></i></a>, <a class="ref" href="msinst_1.html#D4" target="_self" title="MSINST_1:def.4">Def4</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c29">HG</font> being   <a href="msualg_3.html#NM1" title="MSUALG_3:NM.1">ManySortedFunction</a> of <font color="Maroon" title="c21">M2</font>,<font color="Maroon" title="c25">N3</font><b> such that </b><br/><a NAME="E21:13_1_1_2"/><i><font color="Green" title="E20">A17</font></i>: 
( <font color="Maroon" title="c29">HG</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c26">H</font> <a href="pboole.html#K11" title="PBOOLE:func.11">**</a> <font color="Maroon" title="c27">G'</font> &amp; <font color="Maroon" title="c29">HG</font> <a href="msualg_3.html#R1" title="MSUALG_3:pred.1">is_homomorphism</a> <font color="Maroon" title="c21">M2</font>,<font color="Maroon" title="c25">N3</font> )
 <b>by </b><i><a class="txt" href="msinst_1.html#E7:13_1_1_2"><i><font color="Green" title="E12">A9</font></i></a>, <a class="txt" href="msinst_1.html#E9:13_1_1_2"><i><font color="Green" title="E13">A10</font></i></a>, <a class="ref" href="msinst_1.html#T5" target="_self" title="MSINST_1:th.5">Th5</a></i>;<br/>
<a NAME="E22:13_1_1_2"/>
<font color="Maroon" title="c29">HG</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="msinst_1.html#K3" title="MSINST_1:func.3">MSAlg_morph</a> <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c12">j'</font>,<font color="Maroon" title="c14">l'</font>
 
<b>by </b><i><a class="txt" href="msinst_1.html#E7:13_1_1_2"><i><font color="Green" title="E12">A9</font></i></a>, <a class="txt" href="msinst_1.html#E9:13_1_1_2"><i><font color="Green" title="E13">A10</font></i></a>, <a class="txt" href="msinst_1.html#E11:13_1_1_2"><i><font color="Green" title="E15">A12</font></i></a>, <a class="txt" href="msinst_1.html#E21:13_1_1_2"><i><font color="Green" title="E20">A17</font></i></a>, <a class="ref" href="msinst_1.html#D3" target="_self" title="MSINST_1:def.3">Def3</a></i>;<br/>
<a NAME="E23:13_1_1_2"/><b>then </b><i><font color="Green" title="E21">A18</font></i>: 
<font color="Maroon" title="c26">H</font> <a href="pboole.html#K11" title="PBOOLE:func.11">**</a> <font color="Maroon" title="c27">G'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <span class="p1">(<span class="default"><a href="msinst_1.html#K4" title="MSINST_1:func.4">MSAlgCat</a> <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">A</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c8">j</font>,<font color="Maroon" title="c10">l</font>
 
<b>by </b><i><a class="txt" href="msinst_1.html#E21:13_1_1_2"><i><font color="Green" title="E20">A17</font></i></a>, <a class="ref" href="msinst_1.html#D4" target="_self" title="MSINST_1:def.4">Def4</a></i>;<br/>
<a NAME="E24:13_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c26">H</font> <a href="pboole.html#K11" title="PBOOLE:func.11">**</a> <font color="Maroon" title="c27">G'</font></span>)</span> <a href="pboole.html#K11" title="PBOOLE:func.11">**</a> <font color="Maroon" title="c20">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c26">H</font> <a href="pboole.html#K11" title="PBOOLE:func.11">**</a> <span class="p1">(<span class="default"><font color="Maroon" title="c27">G'</font> <a href="pboole.html#K11" title="PBOOLE:func.11">**</a> <font color="Maroon" title="c20">F</font></span>)</span>
 
<b>by </b><i><a class="ref" href="autalg_1.html#T13" title="AUTALG_1:th.13">AUTALG_1:13</a></i>;<br/>
<b>hence </b><a NAME="E25:13_1_1_2"/>
<span class="p1">(<span class="default">the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <span class="p2">(<span class="default"><a href="msinst_1.html#K4" title="MSINST_1:func.4">MSAlgCat</a> <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">A</font></span>)</span> <a href="altcat_1.html#K4" title="ALTCAT_1:func.4">.</a> <font color="Maroon" title="c7">i</font>,<font color="Maroon" title="c9">k</font>,<font color="Maroon" title="c10">l</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c17">h</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default">the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <span class="p3">(<span class="default"><a href="msinst_1.html#K4" title="MSINST_1:func.4">MSAlgCat</a> <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">A</font></span>)</span> <a href="altcat_1.html#K4" title="ALTCAT_1:func.4">.</a> <font color="Maroon" title="c7">i</font>,<font color="Maroon" title="c8">j</font>,<font color="Maroon" title="c9">k</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c16">g</font>,<font color="Maroon" title="c15">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <span class="p2">(<span class="default"><a href="msinst_1.html#K4" title="MSINST_1:func.4">MSAlgCat</a> <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">A</font></span>)</span> <a href="altcat_1.html#K4" title="ALTCAT_1:func.4">.</a> <font color="Maroon" title="c7">i</font>,<font color="Maroon" title="c8">j</font>,<font color="Maroon" title="c10">l</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <span class="p3">(<span class="default"><a href="msinst_1.html#K4" title="MSINST_1:func.4">MSAlgCat</a> <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">A</font></span>)</span> <a href="altcat_1.html#K4" title="ALTCAT_1:func.4">.</a> <font color="Maroon" title="c8">j</font>,<font color="Maroon" title="c9">k</font>,<font color="Maroon" title="c10">l</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c17">h</font>,<font color="Maroon" title="c16">g</font></span>)</span>,<font color="Maroon" title="c15">f</font>
 <b>by </b><i><a class="txt" href="msinst_1.html#E2:13_1_1_2"><i><font color="Green" title="E9">A6</font></i></a>, <a class="txt" href="msinst_1.html#E5:13_1_1_2"><i><font color="Green" title="E11">A8</font></i></a>, <a class="txt" href="msinst_1.html#E7:13_1_1_2"><i><font color="Green" title="E12">A9</font></i></a>, <a class="txt" href="msinst_1.html#E9:13_1_1_2"><i><font color="Green" title="E13">A10</font></i></a>, <a class="txt" href="msinst_1.html#E13:13_1_1_2"><i><font color="Green" title="E16">A13</font></i></a>, <a class="txt" href="msinst_1.html#E15:13_1_1_2"><i><font color="Green" title="E17">A14</font></i></a>, <a class="txt" href="msinst_1.html#E18:13_1_1_2"><i><font color="Green" title="E18">A15</font></i></a>, <a class="txt" href="msinst_1.html#E19:13_1_1_2"><i><font color="Green" title="E19">A16</font></i></a>, <a class="txt" href="msinst_1.html#E23:13_1_1_2"><i><font color="Green" title="E21">A18</font></i></a>, <a class="ref" href="msinst_1.html#D4" target="_self" title="MSINST_1:def.4">Def4</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E3:13_1_1"/>
the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <span class="p1">(<span class="default"><a href="msinst_1.html#K4" title="MSINST_1:func.4">MSAlgCat</a> <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">A</font></span>)</span> is <a href="altcat_1.html#V5" title="ALTCAT_1:attr.5">with_left_units</a>
  <i><font color="Red">:: according to </font></i><a class="ref" href="altcat_1.html#D18" title="ALTCAT_1:def.18">ALTCAT_1:def 18</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <span class="p1">(<span class="default"><a href="msinst_1.html#K4" title="MSINST_1:func.4">MSAlgCat</a> <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">A</font></span>)</span> is <a href="altcat_1.html#V4" title="ALTCAT_1:attr.4">with_right_units</a></span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_1_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c7">j</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="msinst_1.html#K4" title="MSINST_1:func.4">MSAlgCat</a> <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">A</font></span>)</span>; <i><font color="Red">:: according to </font></i><a class="ref" href="altcat_1.html#D9" title="ALTCAT_1:def.9">ALTCAT_1:def 9</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <span class="p1">(<span class="default"><a href="msinst_1.html#K4" title="MSINST_1:func.4">MSAlgCat</a> <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">A</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c7">j</font>,<font color="Maroon" title="c7">j</font> &amp; ( for <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="msinst_1.html#K4" title="MSINST_1:func.4">MSAlgCat</a> <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">A</font></span>)</span><br/> for <font color="Olive">b<sub>3</sub></font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   holds <br/>(  not <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <span class="p1">(<span class="default"><a href="msinst_1.html#K4" title="MSINST_1:func.4">MSAlgCat</a> <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">A</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Olive">b<sub>2</sub></font>,<font color="Maroon" title="c7">j</font> or <span class="p1">(<span class="default">the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <span class="p2">(<span class="default"><a href="msinst_1.html#K4" title="MSINST_1:func.4">MSAlgCat</a> <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">A</font></span>)</span> <a href="altcat_1.html#K4" title="ALTCAT_1:func.4">.</a> <font color="Olive">b<sub>2</sub></font>,<font color="Maroon" title="c7">j</font>,<font color="Maroon" title="c7">j</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive">b<sub>3</sub></font> ) ) )</span><br/>

<b>reconsider </b><font color="Maroon" title="c8">j'</font> = <font color="Maroon" title="c7">j</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="msinst_1.html#K2" title="MSINST_1:func.2">MSAlg_set</a> <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">A</font> <b>by </b><i><a class="ref" href="msinst_1.html#D4" target="_self" title="MSINST_1:def.4">Def4</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c9">MS</font> being  <a href="msualg_1.html#V4" title="MSUALG_1:attr.4">strict</a> <a href="msualg_6.html#V1" title="MSUALG_6:attr.1">feasible</a>  <a href="msualg_1.html#L3" title="MSUALG_1:struct.3">MSAlgebra</a> of <font color="Maroon" title="c1">S</font><b> such that </b><br/><a NAME="E3:13_1_1_3"/><i><font color="Green" title="E9">A19</font></i>: 
( <font color="Maroon" title="c9">MS</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">j'</font> &amp; ( for <font color="Olive" title="b1">C</font> being  <a href="pboole.html#NM3" title="PBOOLE:NM.3">Component</a> of the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c9">MS</font> holds  <font color="Olive" title="b1">C</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">A</font> ) )
 <b>by </b><i><a class="ref" href="msinst_1.html#D2" target="_self" title="MSINST_1:def.2">Def2</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c10">e</font> =  <a href="msualg_3.html#K2" title="MSUALG_3:func.2">id</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c9">MS</font> as   <a href="msualg_3.html#NM1" title="MSUALG_3:NM.1">ManySortedFunction</a> of <font color="Maroon" title="c9">MS</font>,<font color="Maroon" title="c9">MS</font> ;<br/>
<a NAME="E5:13_1_1_3"/><i><font color="Green" title="E10">A20</font></i>: 
<font color="Maroon" title="c10">e</font> <a href="msualg_3.html#R1" title="MSUALG_3:pred.1">is_homomorphism</a> <font color="Maroon" title="c9">MS</font>,<font color="Maroon" title="c9">MS</font>
 
<b>by </b><i><a class="ref" href="msualg_3.html#T9" title="MSUALG_3:th.9">MSUALG_3:9</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c10">e</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c10">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <span class="p1">(<span class="default"><a href="msinst_1.html#K4" title="MSINST_1:func.4">MSAlgCat</a> <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">A</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c7">j</font>,<font color="Maroon" title="c7">j</font> &amp; ( for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="msinst_1.html#K4" title="MSINST_1:func.4">MSAlgCat</a> <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">A</font></span>)</span><br/> for <font color="Olive">b<sub>2</sub></font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   holds <br/>(  not <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <span class="p1">(<span class="default"><a href="msinst_1.html#K4" title="MSINST_1:func.4">MSAlgCat</a> <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">A</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font>,<font color="Maroon" title="c7">j</font> or <span class="p1">(<span class="default">the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <span class="p2">(<span class="default"><a href="msinst_1.html#K4" title="MSINST_1:func.4">MSAlgCat</a> <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">A</font></span>)</span> <a href="altcat_1.html#K4" title="ALTCAT_1:func.4">.</a> <font color="Olive">b<sub>1</sub></font>,<font color="Maroon" title="c7">j</font>,<font color="Maroon" title="c7">j</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c10">e</font>,<font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive">b<sub>2</sub></font> ) ) )</span><br/>

<a NAME="E6:13_1_1_3"/>
the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <span class="p1">(<span class="default"><a href="msinst_1.html#K4" title="MSINST_1:func.4">MSAlgCat</a> <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">A</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c7">j</font>,<font color="Maroon" title="c7">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="msinst_1.html#K3" title="MSINST_1:func.3">MSAlg_morph</a> <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c8">j'</font>,<font color="Maroon" title="c8">j'</font>
 
<b>by </b><i><a class="ref" href="msinst_1.html#D4" target="_self" title="MSINST_1:def.4">Def4</a></i>;<br/>
<b>hence </b><a NAME="E7:13_1_1_3"/><i><font color="Green" title="E11">A21</font></i>: 
<font color="Maroon" title="c10">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <span class="p1">(<span class="default"><a href="msinst_1.html#K4" title="MSINST_1:func.4">MSAlgCat</a> <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">A</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c7">j</font>,<font color="Maroon" title="c7">j</font>
 <b>by </b><i><a class="txt" href="msinst_1.html#E3:13_1_1_3"><i><font color="Green" title="E9">A19</font></i></a>, <a class="txt" href="msinst_1.html#E5:13_1_1_3"><i><font color="Green" title="E10">A20</font></i></a>, <a class="ref" href="msinst_1.html#D3" target="_self" title="MSINST_1:def.3">Def3</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="msinst_1.html#K4" title="MSINST_1:func.4">MSAlgCat</a> <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">A</font></span>)</span><br/> for <font color="Olive">b<sub>2</sub></font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   holds <br/>(  not <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <span class="p1">(<span class="default"><a href="msinst_1.html#K4" title="MSINST_1:func.4">MSAlgCat</a> <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">A</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font>,<font color="Maroon" title="c7">j</font> or <span class="p1">(<span class="default">the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <span class="p2">(<span class="default"><a href="msinst_1.html#K4" title="MSINST_1:func.4">MSAlgCat</a> <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">A</font></span>)</span> <a href="altcat_1.html#K4" title="ALTCAT_1:func.4">.</a> <font color="Olive">b<sub>1</sub></font>,<font color="Maroon" title="c7">j</font>,<font color="Maroon" title="c7">j</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c10">e</font>,<font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive">b<sub>2</sub></font> )</span><br/>

<b>let </b><font color="Maroon" title="c11">i</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="msinst_1.html#K4" title="MSINST_1:func.4">MSAlgCat</a> <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">A</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   holds <br/>(  not <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <span class="p1">(<span class="default"><a href="msinst_1.html#K4" title="MSINST_1:func.4">MSAlgCat</a> <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">A</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c11">i</font>,<font color="Maroon" title="c7">j</font> or <span class="p1">(<span class="default">the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <span class="p2">(<span class="default"><a href="msinst_1.html#K4" title="MSINST_1:func.4">MSAlgCat</a> <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">A</font></span>)</span> <a href="altcat_1.html#K4" title="ALTCAT_1:func.4">.</a> <font color="Maroon" title="c11">i</font>,<font color="Maroon" title="c7">j</font>,<font color="Maroon" title="c7">j</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c10">e</font>,<font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive">b<sub>1</sub></font> )</span><br/><b>let </b><font color="Maroon" title="c12">f</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c12">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <span class="p1">(<span class="default"><a href="msinst_1.html#K4" title="MSINST_1:func.4">MSAlgCat</a> <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">A</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c11">i</font>,<font color="Maroon" title="c7">j</font> or <span class="p1">(<span class="default">the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <span class="p2">(<span class="default"><a href="msinst_1.html#K4" title="MSINST_1:func.4">MSAlgCat</a> <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">A</font></span>)</span> <a href="altcat_1.html#K4" title="ALTCAT_1:func.4">.</a> <font color="Maroon" title="c11">i</font>,<font color="Maroon" title="c7">j</font>,<font color="Maroon" title="c7">j</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c10">e</font>,<font color="Maroon" title="c12">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">f</font> )</span><br/>



<b>reconsider </b><font color="Maroon" title="c13">i'</font> = <font color="Maroon" title="c11">i</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="msinst_1.html#K2" title="MSINST_1:func.2">MSAlg_set</a> <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">A</font> <b>by </b><i><a class="ref" href="msinst_1.html#D4" target="_self" title="MSINST_1:def.4">Def4</a></i>;<br/>
<b>assume </b><a NAME="E9:13_1_1_3"/><i><font color="Green" title="E12">A22</font></i>: 
<font color="Maroon" title="c12">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <span class="p1">(<span class="default"><a href="msinst_1.html#K4" title="MSINST_1:func.4">MSAlgCat</a> <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">A</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c11">i</font>,<font color="Maroon" title="c7">j</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default">the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <span class="p2">(<span class="default"><a href="msinst_1.html#K4" title="MSINST_1:func.4">MSAlgCat</a> <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">A</font></span>)</span> <a href="altcat_1.html#K4" title="ALTCAT_1:func.4">.</a> <font color="Maroon" title="c11">i</font>,<font color="Maroon" title="c7">j</font>,<font color="Maroon" title="c7">j</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c10">e</font>,<font color="Maroon" title="c12">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">f</font></span><br/>

<a NAME="E10:13_1_1_3"/><b>then </b>
<font color="Maroon" title="c12">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="msinst_1.html#K3" title="MSINST_1:func.3">MSAlg_morph</a> <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c13">i'</font>,<font color="Maroon" title="c8">j'</font>
 
<b>by </b><i><a class="ref" href="msinst_1.html#D4" target="_self" title="MSINST_1:def.4">Def4</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c14">M1</font>, <font color="Maroon" title="c15">N1</font> being  <a href="msualg_1.html#V4" title="MSUALG_1:attr.4">strict</a> <a href="msualg_6.html#V1" title="MSUALG_6:attr.1">feasible</a>  <a href="msualg_1.html#L3" title="MSUALG_1:struct.3">MSAlgebra</a> of <font color="Maroon" title="c1">S</font>, <font color="Maroon" title="c16">F</font> being   <a href="msualg_3.html#NM1" title="MSUALG_3:NM.1">ManySortedFunction</a> of <font color="Maroon" title="c14">M1</font>,<font color="Maroon" title="c15">N1</font><b> such that </b><br/><a NAME="E12:13_1_1_3"/><i><font color="Green" title="E13">A23</font></i>: 
( <font color="Maroon" title="c14">M1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">i'</font> &amp; <font color="Maroon" title="c15">N1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">j'</font> &amp; <font color="Maroon" title="c16">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">f</font> &amp; the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c14">M1</font> <a href="pzfmisc1.html#R1" title="PZFMISC1:pred.1">is_transformable_to</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c15">N1</font> &amp; <font color="Maroon" title="c16">F</font> <a href="msualg_3.html#R1" title="MSUALG_3:pred.1">is_homomorphism</a> <font color="Maroon" title="c14">M1</font>,<font color="Maroon" title="c15">N1</font> )
 <b>by </b><i><a class="ref" href="msinst_1.html#D3" target="_self" title="MSINST_1:def.3">Def3</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c17">F</font> = <font color="Maroon" title="c16">F</font> as   <a href="msualg_3.html#NM1" title="MSUALG_3:NM.1">ManySortedFunction</a> of <font color="Maroon" title="c14">M1</font>,<font color="Maroon" title="c9">MS</font> <b>by </b><i><a class="txt" href="msinst_1.html#E3:13_1_1_3"><i><font color="Green" title="E9">A19</font></i></a>, <a class="txt" href="msinst_1.html#E12:13_1_1_3"><i><font color="Green" title="E13">A23</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c18">I</font> = <font color="Maroon" title="c11">i</font>, <font color="Maroon" title="c19">J</font> = <font color="Maroon" title="c7">j</font> as   <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <span class="p1">(<span class="default"><a href="msinst_1.html#K4" title="MSINST_1:func.4">MSAlgCat</a> <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">A</font></span>)</span> ;<br/>
<a NAME="E15:13_1_1_3"/>
<span class="p1">(<span class="default">the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <span class="p2">(<span class="default"><a href="msinst_1.html#K4" title="MSINST_1:func.4">MSAlgCat</a> <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">A</font></span>)</span> <a href="altcat_1.html#K4" title="ALTCAT_1:func.4">.</a> <font color="Maroon" title="c18">I</font>,<font color="Maroon" title="c19">J</font>,<font color="Maroon" title="c19">J</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c10">e</font>,<font color="Maroon" title="c12">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">e</font> <a href="pboole.html#K11" title="PBOOLE:func.11">**</a> <font color="Maroon" title="c17">F</font>
 
<b>by </b><i><a class="txt" href="msinst_1.html#E7:13_1_1_3"><i><font color="Green" title="E11">A21</font></i></a>, <a class="txt" href="msinst_1.html#E9:13_1_1_3"><i><font color="Green" title="E12">A22</font></i></a>, <a class="txt" href="msinst_1.html#E12:13_1_1_3"><i><font color="Green" title="E13">A23</font></i></a>, <a class="ref" href="msinst_1.html#D4" target="_self" title="MSINST_1:def.4">Def4</a></i>;<br/>
<b>hence </b><a NAME="E16:13_1_1_3"/>
<span class="p1">(<span class="default">the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <span class="p2">(<span class="default"><a href="msinst_1.html#K4" title="MSINST_1:func.4">MSAlgCat</a> <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">A</font></span>)</span> <a href="altcat_1.html#K4" title="ALTCAT_1:func.4">.</a> <font color="Maroon" title="c11">i</font>,<font color="Maroon" title="c7">j</font>,<font color="Maroon" title="c7">j</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c10">e</font>,<font color="Maroon" title="c12">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">f</font>
 <b>by </b><i><a class="txt" href="msinst_1.html#E12:13_1_1_3"><i><font color="Green" title="E13">A23</font></i></a>, <a class="ref" href="msualg_3.html#T4" title="MSUALG_3:th.4">MSUALG_3:4</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E4:13_1_1"/>
the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <span class="p1">(<span class="default"><a href="msinst_1.html#K4" title="MSINST_1:func.4">MSAlgCat</a> <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">A</font></span>)</span> is <a href="altcat_1.html#V4" title="ALTCAT_1:attr.4">with_right_units</a>
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_1_1_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c7">i</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="msinst_1.html#K4" title="MSINST_1:func.4">MSAlgCat</a> <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">A</font></span>)</span>; <i><font color="Red">:: according to </font></i><a class="ref" href="altcat_1.html#D8" title="ALTCAT_1:def.8">ALTCAT_1:def 8</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <span class="p1">(<span class="default"><a href="msinst_1.html#K4" title="MSINST_1:func.4">MSAlgCat</a> <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">A</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c7">i</font>,<font color="Maroon" title="c7">i</font> &amp; ( for <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="msinst_1.html#K4" title="MSINST_1:func.4">MSAlgCat</a> <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">A</font></span>)</span><br/> for <font color="Olive">b<sub>3</sub></font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   holds <br/>(  not <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <span class="p1">(<span class="default"><a href="msinst_1.html#K4" title="MSINST_1:func.4">MSAlgCat</a> <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">A</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c7">i</font>,<font color="Olive">b<sub>2</sub></font> or <span class="p1">(<span class="default">the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <span class="p2">(<span class="default"><a href="msinst_1.html#K4" title="MSINST_1:func.4">MSAlgCat</a> <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">A</font></span>)</span> <a href="altcat_1.html#K4" title="ALTCAT_1:func.4">.</a> <font color="Maroon" title="c7">i</font>,<font color="Maroon" title="c7">i</font>,<font color="Olive">b<sub>2</sub></font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Olive">b<sub>3</sub></font>,<font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive">b<sub>3</sub></font> ) ) )</span><br/>

<b>reconsider </b><font color="Maroon" title="c8">i'</font> = <font color="Maroon" title="c7">i</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="msinst_1.html#K2" title="MSINST_1:func.2">MSAlg_set</a> <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">A</font> <b>by </b><i><a class="ref" href="msinst_1.html#D4" target="_self" title="MSINST_1:def.4">Def4</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c9">MS</font> being  <a href="msualg_1.html#V4" title="MSUALG_1:attr.4">strict</a> <a href="msualg_6.html#V1" title="MSUALG_6:attr.1">feasible</a>  <a href="msualg_1.html#L3" title="MSUALG_1:struct.3">MSAlgebra</a> of <font color="Maroon" title="c1">S</font><b> such that </b><br/><a NAME="E3:13_1_1_4"/><i><font color="Green" title="E9">A24</font></i>: 
( <font color="Maroon" title="c9">MS</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">i'</font> &amp; ( for <font color="Olive" title="b1">C</font> being  <a href="pboole.html#NM3" title="PBOOLE:NM.3">Component</a> of the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c9">MS</font> holds  <font color="Olive" title="b1">C</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">A</font> ) )
 <b>by </b><i><a class="ref" href="msinst_1.html#D2" target="_self" title="MSINST_1:def.2">Def2</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c10">e</font> =  <a href="msualg_3.html#K2" title="MSUALG_3:func.2">id</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c9">MS</font> as   <a href="msualg_3.html#NM1" title="MSUALG_3:NM.1">ManySortedFunction</a> of <font color="Maroon" title="c9">MS</font>,<font color="Maroon" title="c9">MS</font> ;<br/>
<a NAME="E5:13_1_1_4"/><i><font color="Green" title="E10">A25</font></i>: 
<font color="Maroon" title="c10">e</font> <a href="msualg_3.html#R1" title="MSUALG_3:pred.1">is_homomorphism</a> <font color="Maroon" title="c9">MS</font>,<font color="Maroon" title="c9">MS</font>
 
<b>by </b><i><a class="ref" href="msualg_3.html#T9" title="MSUALG_3:th.9">MSUALG_3:9</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c10">e</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c10">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <span class="p1">(<span class="default"><a href="msinst_1.html#K4" title="MSINST_1:func.4">MSAlgCat</a> <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">A</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c7">i</font>,<font color="Maroon" title="c7">i</font> &amp; ( for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="msinst_1.html#K4" title="MSINST_1:func.4">MSAlgCat</a> <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">A</font></span>)</span><br/> for <font color="Olive">b<sub>2</sub></font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   holds <br/>(  not <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <span class="p1">(<span class="default"><a href="msinst_1.html#K4" title="MSINST_1:func.4">MSAlgCat</a> <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">A</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c7">i</font>,<font color="Olive">b<sub>1</sub></font> or <span class="p1">(<span class="default">the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <span class="p2">(<span class="default"><a href="msinst_1.html#K4" title="MSINST_1:func.4">MSAlgCat</a> <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">A</font></span>)</span> <a href="altcat_1.html#K4" title="ALTCAT_1:func.4">.</a> <font color="Maroon" title="c7">i</font>,<font color="Maroon" title="c7">i</font>,<font color="Olive">b<sub>1</sub></font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Olive">b<sub>2</sub></font>,<font color="Maroon" title="c10">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive">b<sub>2</sub></font> ) ) )</span><br/>

<a NAME="E6:13_1_1_4"/>
the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <span class="p1">(<span class="default"><a href="msinst_1.html#K4" title="MSINST_1:func.4">MSAlgCat</a> <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">A</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c7">i</font>,<font color="Maroon" title="c7">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="msinst_1.html#K3" title="MSINST_1:func.3">MSAlg_morph</a> <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c8">i'</font>,<font color="Maroon" title="c8">i'</font>
 
<b>by </b><i><a class="ref" href="msinst_1.html#D4" target="_self" title="MSINST_1:def.4">Def4</a></i>;<br/>
<b>hence </b><a NAME="E7:13_1_1_4"/><i><font color="Green" title="E11">A26</font></i>: 
<font color="Maroon" title="c10">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <span class="p1">(<span class="default"><a href="msinst_1.html#K4" title="MSINST_1:func.4">MSAlgCat</a> <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">A</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c7">i</font>,<font color="Maroon" title="c7">i</font>
 <b>by </b><i><a class="txt" href="msinst_1.html#E3:13_1_1_4"><i><font color="Green" title="E9">A24</font></i></a>, <a class="txt" href="msinst_1.html#E5:13_1_1_4"><i><font color="Green" title="E10">A25</font></i></a>, <a class="ref" href="msinst_1.html#D3" target="_self" title="MSINST_1:def.3">Def3</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="msinst_1.html#K4" title="MSINST_1:func.4">MSAlgCat</a> <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">A</font></span>)</span><br/> for <font color="Olive">b<sub>2</sub></font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   holds <br/>(  not <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <span class="p1">(<span class="default"><a href="msinst_1.html#K4" title="MSINST_1:func.4">MSAlgCat</a> <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">A</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c7">i</font>,<font color="Olive">b<sub>1</sub></font> or <span class="p1">(<span class="default">the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <span class="p2">(<span class="default"><a href="msinst_1.html#K4" title="MSINST_1:func.4">MSAlgCat</a> <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">A</font></span>)</span> <a href="altcat_1.html#K4" title="ALTCAT_1:func.4">.</a> <font color="Maroon" title="c7">i</font>,<font color="Maroon" title="c7">i</font>,<font color="Olive">b<sub>1</sub></font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Olive">b<sub>2</sub></font>,<font color="Maroon" title="c10">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive">b<sub>2</sub></font> )</span><br/>

<b>let </b><font color="Maroon" title="c11">j</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="msinst_1.html#K4" title="MSINST_1:func.4">MSAlgCat</a> <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">A</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   holds <br/>(  not <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <span class="p1">(<span class="default"><a href="msinst_1.html#K4" title="MSINST_1:func.4">MSAlgCat</a> <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">A</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c7">i</font>,<font color="Maroon" title="c11">j</font> or <span class="p1">(<span class="default">the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <span class="p2">(<span class="default"><a href="msinst_1.html#K4" title="MSINST_1:func.4">MSAlgCat</a> <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">A</font></span>)</span> <a href="altcat_1.html#K4" title="ALTCAT_1:func.4">.</a> <font color="Maroon" title="c7">i</font>,<font color="Maroon" title="c7">i</font>,<font color="Maroon" title="c11">j</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font>,<font color="Maroon" title="c10">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive">b<sub>1</sub></font> )</span><br/><b>let </b><font color="Maroon" title="c12">f</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c12">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <span class="p1">(<span class="default"><a href="msinst_1.html#K4" title="MSINST_1:func.4">MSAlgCat</a> <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">A</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c7">i</font>,<font color="Maroon" title="c11">j</font> or <span class="p1">(<span class="default">the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <span class="p2">(<span class="default"><a href="msinst_1.html#K4" title="MSINST_1:func.4">MSAlgCat</a> <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">A</font></span>)</span> <a href="altcat_1.html#K4" title="ALTCAT_1:func.4">.</a> <font color="Maroon" title="c7">i</font>,<font color="Maroon" title="c7">i</font>,<font color="Maroon" title="c11">j</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c12">f</font>,<font color="Maroon" title="c10">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">f</font> )</span><br/>



<b>reconsider </b><font color="Maroon" title="c13">j'</font> = <font color="Maroon" title="c11">j</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="msinst_1.html#K2" title="MSINST_1:func.2">MSAlg_set</a> <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">A</font> <b>by </b><i><a class="ref" href="msinst_1.html#D4" target="_self" title="MSINST_1:def.4">Def4</a></i>;<br/>
<b>assume </b><a NAME="E9:13_1_1_4"/><i><font color="Green" title="E12">A27</font></i>: 
<font color="Maroon" title="c12">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <span class="p1">(<span class="default"><a href="msinst_1.html#K4" title="MSINST_1:func.4">MSAlgCat</a> <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">A</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c7">i</font>,<font color="Maroon" title="c11">j</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default">the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <span class="p2">(<span class="default"><a href="msinst_1.html#K4" title="MSINST_1:func.4">MSAlgCat</a> <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">A</font></span>)</span> <a href="altcat_1.html#K4" title="ALTCAT_1:func.4">.</a> <font color="Maroon" title="c7">i</font>,<font color="Maroon" title="c7">i</font>,<font color="Maroon" title="c11">j</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c12">f</font>,<font color="Maroon" title="c10">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">f</font></span><br/>

<a NAME="E10:13_1_1_4"/><b>then </b>
<font color="Maroon" title="c12">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="msinst_1.html#K3" title="MSINST_1:func.3">MSAlg_morph</a> <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c8">i'</font>,<font color="Maroon" title="c13">j'</font>
 
<b>by </b><i><a class="ref" href="msinst_1.html#D4" target="_self" title="MSINST_1:def.4">Def4</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c14">M1</font>, <font color="Maroon" title="c15">N1</font> being  <a href="msualg_1.html#V4" title="MSUALG_1:attr.4">strict</a> <a href="msualg_6.html#V1" title="MSUALG_6:attr.1">feasible</a>  <a href="msualg_1.html#L3" title="MSUALG_1:struct.3">MSAlgebra</a> of <font color="Maroon" title="c1">S</font>, <font color="Maroon" title="c16">F</font> being   <a href="msualg_3.html#NM1" title="MSUALG_3:NM.1">ManySortedFunction</a> of <font color="Maroon" title="c14">M1</font>,<font color="Maroon" title="c15">N1</font><b> such that </b><br/><a NAME="E12:13_1_1_4"/><i><font color="Green" title="E13">A28</font></i>: 
( <font color="Maroon" title="c14">M1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">i'</font> &amp; <font color="Maroon" title="c15">N1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">j'</font> &amp; <font color="Maroon" title="c16">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">f</font> &amp; the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c14">M1</font> <a href="pzfmisc1.html#R1" title="PZFMISC1:pred.1">is_transformable_to</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c15">N1</font> &amp; <font color="Maroon" title="c16">F</font> <a href="msualg_3.html#R1" title="MSUALG_3:pred.1">is_homomorphism</a> <font color="Maroon" title="c14">M1</font>,<font color="Maroon" title="c15">N1</font> )
 <b>by </b><i><a class="ref" href="msinst_1.html#D3" target="_self" title="MSINST_1:def.3">Def3</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c17">F</font> = <font color="Maroon" title="c16">F</font> as   <a href="msualg_3.html#NM1" title="MSUALG_3:NM.1">ManySortedFunction</a> of <font color="Maroon" title="c9">MS</font>,<font color="Maroon" title="c15">N1</font> <b>by </b><i><a class="txt" href="msinst_1.html#E3:13_1_1_4"><i><font color="Green" title="E9">A24</font></i></a>, <a class="txt" href="msinst_1.html#E12:13_1_1_4"><i><font color="Green" title="E13">A28</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c18">I</font> = <font color="Maroon" title="c7">i</font>, <font color="Maroon" title="c19">J</font> = <font color="Maroon" title="c11">j</font> as   <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <span class="p1">(<span class="default"><a href="msinst_1.html#K4" title="MSINST_1:func.4">MSAlgCat</a> <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">A</font></span>)</span> ;<br/>
<a NAME="E15:13_1_1_4"/>
<span class="p1">(<span class="default">the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <span class="p2">(<span class="default"><a href="msinst_1.html#K4" title="MSINST_1:func.4">MSAlgCat</a> <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">A</font></span>)</span> <a href="altcat_1.html#K4" title="ALTCAT_1:func.4">.</a> <font color="Maroon" title="c18">I</font>,<font color="Maroon" title="c18">I</font>,<font color="Maroon" title="c19">J</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c12">f</font>,<font color="Maroon" title="c10">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">F</font> <a href="pboole.html#K11" title="PBOOLE:func.11">**</a> <font color="Maroon" title="c10">e</font>
 
<b>by </b><i><a class="txt" href="msinst_1.html#E7:13_1_1_4"><i><font color="Green" title="E11">A26</font></i></a>, <a class="txt" href="msinst_1.html#E9:13_1_1_4"><i><font color="Green" title="E12">A27</font></i></a>, <a class="txt" href="msinst_1.html#E12:13_1_1_4"><i><font color="Green" title="E13">A28</font></i></a>, <a class="ref" href="msinst_1.html#D4" target="_self" title="MSINST_1:def.4">Def4</a></i>;<br/>
<b>hence </b><a NAME="E16:13_1_1_4"/>
<span class="p1">(<span class="default">the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <span class="p2">(<span class="default"><a href="msinst_1.html#K4" title="MSINST_1:func.4">MSAlgCat</a> <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">A</font></span>)</span> <a href="altcat_1.html#K4" title="ALTCAT_1:func.4">.</a> <font color="Maroon" title="c7">i</font>,<font color="Maroon" title="c7">i</font>,<font color="Maroon" title="c11">j</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c12">f</font>,<font color="Maroon" title="c10">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">f</font>
 <b>by </b><i><a class="txt" href="msinst_1.html#E12:13_1_1_4"><i><font color="Green" title="E13">A28</font></i></a>, <a class="ref" href="msualg_3.html#T3" title="MSUALG_3:th.3">MSUALG_3:3</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>


</div>
